By Claude E. Shannon
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This is often the vintage introductory graduate textual content. center of the e-book is degree thought and Lebesque integration.
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Additional resources for A Mathematical Theory of Communication
S C W log 1 + N ! 1: S We wish to maximize the entropy of the received signal. If is large this will occur very nearly when N we maximize the entropy of the transmitted ensemble. The asymptotic upper bound is obtained by relaxing the conditions on the ensemble. Let us suppose that the power is limited to S not at every instant of time, but only at the sample points. The maximum entropy of the transmitted ensemble under these weakened conditions is certainly greater than or equal to that under the original conditions.
The Jacobian of the transformation is (for n sine and n cosine components) n J = ∏ jY fi j2 i=1 where the fi are equally spaced through the band W . This becomes in the limit exp 1 W Z W log jY f j2 d f : Since J is constant its average value is the same quantity and applying the theorem on the change of entropy with a change of coordinates, the result follows. We may also phrase it in terms of the entropy power. Thus if the entropy power of the first ensemble is N1 that of the second is N1 exp 1 W Z W log jY f j2 d f 39 : TABLE I ENTROPY ENTROPY POWER POWER GAIN FACTOR IN DECIBELS GAIN IMPULSE RESPONSE 1 1,!
3 ! 2 2 ,2 67 2 e ! 0 : J1 t 2 t 1 1 ,8 69 1 e2 ! 0 : 1 cos1 , t2 t , cost 1 The final entropy power is the initial entropy power multiplied by the geometric mean gain of the filter. If the gain is measured in db, then the output entropy power will be increased by the arithmetic mean db gain over W . In Table I the entropy power loss has been calculated (and also expressed in db) for a number of ideal gain characteristics. The impulsive responses of these filters are also given for W = 2 , with phase assumed to be 0.